Summary: Problem Set 2
Problem 1. a) Let I = (x2
 1) be an ideal in R[x]. Is I maximal? Why or why
not?
b) Let I = (x2
+ 1) be an ideal in R[x]. Is I maximal? Why or why not?
c) Let I = (x2
+ 1) be an ideal in C[x]. Is I maximal? Why or why not?
Problem 2. a) Let I be an ideal in a ring R. Suppose fn
I and gm
I. Show
that (f + g)n+m
I.
b) Let J = {f Rft
I for some t > 0}. Show that J is an ideal.
Remark 3. The ideal J dened in the problem above is a radical ideal and is the
smallest radical ideal which contains I. Hence J = rad(I). Many books dene
the radical in this way i.e. they dene rad(I) = {f Rft
I for some t >
0}. It is equivalent to the denition given in the handout entitled "Some Useful
